Beginning Algebra With Applications, Chapter 7, 7.1, Section 7.1, Problem 66

What polynomial must be subtracted from $2x^3 - x^2 + 4x - 2$ so that the difference is $x^3 + 2x - 8$?

Let $P$ = the polynomial that must be subtracted


$
\begin{equation}
\begin{aligned}

& 2x^3 - x^2 + 4x - 2 - P = x^3 + 2x - 8
&&
\\
& (2x^3 - x^2 + 4x - 2) - (x^3 + 2x - 8) = P
&&
\\
& (2x^3 - x^3) - x^2 + (4x-2x) + (-2+8) = P
&& \text{Rewrite subtraction as addition of the opposite}
\\
& x^3 - x^2 + 2x + 6 = P
&& \text{Combine like terms and write the polynomial in descending order.}


\end{aligned}
\end{equation}
$